How was trigonometric functions and logarithms calculated?

[hobz]

I was wondering how the ancient Greeks calculated sine, cosine, etc.

My guess is that they measured, and put the values in a table.

 

Likewise, how was the first logarithms (or inverse functions in general) calculated?

[Izzy_Bee]

Maybe they found a need for it so they answered it by creating new names.. For example, if you had the numbers 1, 2, 3, 6, 7 you would wonder is there a number in between (this is if numbers where unknown) And you could say I decide that It should go 1, 2, 3, 4, 5, 6, 7 etc etc. Or if you have A, B, C, F, G you could say no I think it should go A, B, C, D, E, F, G etc etc. New things are found everyday and need new names the founder will name them. I assume you know what sine and cisone are so I dont need to explain that (:

I hope this has helped.

Also- If you have a time and a distance you can work out the speed you would have to go, If you have speen and time you can work out the distance you can travel.. So i think they had maybe sine but there was something missing so they come up with cosine,, This is just a idea.

Edited July 7, 2010 by TRSNZIREN1
Missed Information

[hobz]

I agree, but I think you misunderstood my question. I am asking how these things were calculated. Not how the need for them (or their names) came to be.

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(Sorry for the "How was", but I appended logarithms at the last moment, and forgot to change to "were")

[the tree]

For a large part, I think measured by hand. Although the 'nice' values for the trig functions can be proven from the geometric axioms.

 

Logarithms only came up in the past couple of centuries, they'd have been done by (and I'd imagine still are to an extent) by approximate Riemann sums.

[hobz]

For a large part, I think measured by hand. Although the 'nice' values for the trig functions can be proven from the geometric axioms.

 

Logarithms only came up in the past couple of centuries, they'd have been done by (and I'd imagine still are to an extent) by approximate Riemann sums.

Interesting. Logarithms are far older than Riemann sums, as far as I know.

 

the calculator requires the use of javascript enabled and capable browsers This script will calculate any combination of logarithms, trigonometric functions, and the logarithms of trigonometric functions. Enter the angle value you want to apply the function to in decimal degrees.

Although the ancient Greeks are generally credited for alot of things, I doubt that javascript is one of them.

[the tree]

Interesting. Logarithms are far older than Riemann sums, as far as I know.

I didn't mean particularly formal Riemann sums, just that it'd have been done by adding up values of 1/x.

 

edit: oh and I'm pretty sure Styla is a spambot, which would explain the nonsense, I've reported the post.

Edited July 9, 2010 by the tree

[GeneHerr]

I dimly remember that you can calculate Logarithms using Power Series, but do not recall exactly how this is accomplished.

[baxtrom]

Newton et al used series expansions to calculate logarithms. For example like this. And, if you've calculated logarithms of numbers from say 1 to 100, you can obtain log(500) as log(5) + 2*log(10), and log(13.2) as log(66/5) = log(66) - log(5) and so on. In used books stores you can still find old books with tables loaded with logarithms, probably calculated by some poor PhD student.

[Guest sofiarunner]

Guys zero was invented by an Indian, then how come Greeks calculated sin90???

[baxtrom]

Guys zero was invented by an Indian, then how come Greeks calculated sin90???

 

Because sin(90 deg) = 1?

[michel123456]

I was wondering how the ancient Greeks calculated sine, cosine, etc.

 

My guess is that they measured, and put the values in a table.

 

 

Likewise, how was the first logarithms (or inverse functions in general) calculated?

 

Here is a concise presentation of Ptolemy's Table of Chords.

--------------------

From the Μεγιστη

 

 

This is a table from the Ptolemy's Almagest as translated in Arabic. The picture is stolen from a presentation in greek page 39, I was not able to find the original picture. (maybe copyright issue?)

Edited May 16, 2011 by michel123456

[baxtrom]

(maybe copyright issue?)

 

 

Nah, I don't think Ptolemy would mind.

[michel123456]

Nah, I don't think Ptolemy would mind.

 

This is derailing a little bit:

 

Sometimes copyright is upon the picture, not upon the original author (which is some Arab mathematician, since the original Ptolemy's work in Greek has been lost).

I see that in Architecture, where photographers put copyright upon pictures of buildings made by others.

[baric]

Guys zero was invented by an Indian, then how come Greeks calculated sin90???

 

The Babylonians had a positional notation, so they understood the concept of zero very early on. Their mathematics predated that of the Greeks by many centuries.

[csmyth3025]

Interesting. Logarithms are far older than Riemann sums, as far as I know.

 

styla786 said:the calculator requires the use of javascript enabled and capable browsers This script will calculate any combination of logarithms, trigonometric functions, and the logarithms of trigonometric functions. Enter the angle value you want to apply the function to in decimal degrees.

Although the ancient Greeks are generally credited for alot of things, I doubt that javascript is one of them.

I think Mr. Charles Babbage and his very able and talented translator (and, as it turns out, first of all programmers) Ms. Ada Agusta, Countess of Agusta preceded javascript by about 150 years. In regard to the automatic calculation of logarithms and trigonometric functions, Ada has this to say:

 

It is well known that the French government, wishing to promote the extension of the decimal system, had ordered the construction of logarithmical and trigonometrical tables of enormous extent. M. de Prony, who had been entrusted with the direction of this undertaking, divided it into three sections, to each of which was appointed a special class of persons. In the first section the formulæ were so combined as to render them subservient to the purposes of numerical calculation; in the second, these same formulæ were calculated for values of the variable, selected at certain successive distances; and under the third section, comprising about eighty individuals, who were most of them only acquainted with the first two rules of arithmetic, the values which were intermediate to those calculated by the second section were interpolated by means of simple additions and subtractions...

 

...An undertaking similar to that just mentioned having been entered upon in England, Mr. Babbage conceived that the operations performed under the third section might be executed by a machine; and this idea he realized by means of mechanism, which has been in part put together, and to which the name Difference Engine is applicable, on account of the principle upon which its construction is founded...

 

Mr Babbage's Anylitical Engine (never built) is an even more versatile and sophisticated machine which, with the input of the remarkable Countess, is believed to have incorporated every fundamental aspect of electronic computing used to this very day.

(ref. http://www.fourmilab...age/sketch.html )

 

In short, before the advent of the successors to the (unfinished) Difference Engine and the (unbuilt) Anylitical engine, logarithms and trigonometric functions were laboriously calculated by hand with pen and paper usng formulas that were even at that time more than a hundred years old.

 

Javascript is a latecomer to the game.

 

Chris

[khaled]

Here is a concise presentation of Ptolemy's Table of Chords.

--------------------

From the Μεγιστη

 

 

This is a table from the Ptolemy's Almagest as translated in Arabic. The picture is stolen from a presentation in greek page 39, I was not able to find the original picture. (maybe copyright issue?)

 

The sheet contains information about celestial horoscopes, such as Aries, Taurus, Gemini, ..etc

 

this sounds weird, so they knew how to calculate angles for the zodiac system

 

Also, about how ancients calculated sin(90), and zero was invented by indian scientist .. I think if you think about it,

Greek have their representation of numbers .. I = 1, V = 5, & X = 10 .. so, sin(90) in greek is simply sin(XXXXXXXXX) = I

.. also, I think they have a symbol for zero (nothing), and based on how they represent the numbers, I think zero

in greek is simply a gap .. 0 = __, 1 = I, 2 = II, 3 = III, ..etc

Edited July 24, 2011 by khaled

[baxtrom]

I think if you think about it, Greek have their representation of numbers .. I = 1, V = 5, & X = 10 .. so, sin(90) in greek is simply sin(XXXXXXXXX) = I

.. also, I think they have a symbol for zero (nothing), and based on how they represent the numbers, I think zero

in greek is simply a gap .. 0 = __, 1 = I, 2 = II, 3 = III, ..etc

 

Two minor comments,

first, I believe you are referring to Roman numerals, not Greek. Second, 90 would be written XC, or "ten from one hundred".

 

Also, good to know Swenglish memory trick when dealing with Roman numerals: Little Camilla's Dancing Mazurka